Deformed semicircle law and concentration of nonlinear random matrices for ultra-wide neural networks

In this paper, we study the two-layer fully connected neural network given by $f(X)=\frac{1}{\sqrt{d_1}}\boldsymbol{a}^\top\sigma\left(WX\right)$, where $X\in\mathbb{R}^{d_0\times n}$ is a deterministic data matrix, $W\in\mathbb{R}^{d_1\times d_0}$ and $\boldsymbol{a}\in\mathbb{R}^{d_1}$ are random Gaussian weights, and $\sigma$ is a nonlinear activation function. We obtain the limiting spectral distributions of two kernel matrices related to $f(X)$: the empirical conjugate kernel (CK) and neural tangent kernel (NTK), beyond the linear-width regime ($d_1\asymp n$). Under the ultra-width regime $d_1/n\to\infty$, with proper assumptions on $X$ and $\sigma$, a deformed semicircle law appears. Such limiting law is first proved for general centered sample covariance matrices with correlation and then specified for our neural network model. We also prove non-asymptotic concentrations of empirical CK and NTK around their limiting kernel in the spectral norm, and lower bounds on their smallest eigenvalues. As an application, we verify the random feature regression achieves the same asymptotic performance as its limiting kernel regression in ultra-width limit. The limiting training and test errors for random feature regression are calculated by corresponding kernel regression. We also provide a nonlinear Hanson-Wright inequality suitable for neural networks with random weights and Lipschitz activation functions.